Japanese Language Stack Exchange is a question and answer site for students, teachers, and linguists wanting to discuss the finer points of the Japanese language. It's 100% free, no registration required.

2 Answers
2

It's worth mentioning that 閉じる has a spatial nuance to it. When I studied Japanese as an undergraduate, we were taught to differentiate 閉める and 閉じる by this nuance. For example:

プログラム、目、扉 ー ひらく、とじる

瓶、パッケージ、ドア ー あける、しめる

What we were taught is that if it involves opening something outward/to the side (and vice versa if it were being closed), like a book or your eyelids, you use 開く・閉じる.

Now this is not a hard and fast rule, but it is enough to illustrate the idea that when you close a door with 閉じる, there is a nuance of pulling it shut. The image for me is a double door or a sliding screen, which, when standing in the doorway, must necessarily be pulled closed, and conveniently, in the case of the sliding screen, pulled open.

This might explain why there is no entry for 押し閉める, and why my IME doesn't recognize it. There's no essential need for it.

There are no dictionary entries for 押し閉める (while there are for 押し開ける), but you can find it used on Google to some extent. There are no entries for it on alc either, though. If you use this term you will probably be understood with no problem but it does't appear to be used very commonly.

As you've added 引き閉める I'll note that I can't find any usages of that, and that it will probably be confused with 引き締める.

A note on suffixes in Japanese: while there are many words that you can mix and match to create unique meanings, a lot of them are only really used in certain combinations and are basically established as words in their own right with their own nuances and usage and everything. Mixing unconventionally may be understandable but it won't always sound "normal"